An efficient semismooth Newton-AMG-based inexact primal-dual algorithm for generalized transport problems

TL;DR

This paper introduces an efficient inexact primal-dual algorithm for large-scale optimal mass transport problems, utilizing semismooth Newton methods and AMG solvers to achieve fast convergence and computational efficiency.

Contribution

It develops a novel inexact primal-dual algorithm based on dynamical systems, with a semismooth Newton approach and AMG solver for the linear systems, improving efficiency for large problems.

Findings

Global super-linear convergence rate established.

Linear systems are effectively solved using AMG methods.

Numerical experiments confirm the method's efficiency.

Abstract

This work is concerned with the efficient optimization method for solving a large class of optimal mass transport problems. An inexact primal-dual algorithm is presented from the time discretization of a proper dynamical system, and by using the tool of Lyapunov function, the global (super-)linear convergence rate is established for function residual and feasibility violation. The proposed algorithm contains an inner problem that possesses strong semismoothness property and motivates the use of the semismooth Newton iteration. By exploring the hidden structure of the problem itself, the linear system arising from the Newton iteration is transferred equivalently into a graph Laplacian system, for which a robust algebraic multigrid method is proposed and also analyzed via the famous Xu--Zikatanov identity. Finally, numerical experiments are provided to validate the efficiency of our method.